To determine how many years it will take for an initial amount of [tex]$9,000 to grow to $[/tex]17,000 with an annual rate of change of 4.647541%, we need to solve using the compound interest formula. Here's a step-by-step solution:
1. Identify Key Variables:
- Initial amount ([tex]\( P \)[/tex]) = [tex]$9,000
- Final amount (\( A \)) = $[/tex]17,000
- Annual rate of change ([tex]\( r \)[/tex]) = 4.647541%
2. Convert the Percentage Rate to Decimal Form:
[tex]\[
r = \frac{4.647541}{100} = 0.04647541
\][/tex]
3. Write Down the Compound Interest Formula to Find Time:
The compound interest formula is given by:
[tex]\[
A = P(1 + r)^t
\][/tex]
Where:
- [tex]\( A \)[/tex] is the final amount
- [tex]\( P \)[/tex] is the initial amount
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form)
- [tex]\( t \)[/tex] is the number of years
4. Rearrange the Formula to Solve for [tex]\( t \)[/tex]:
We need to solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(\frac{A}{P})}{\ln(1 + r)}
\][/tex]
5. Plug in the Values:
- [tex]\( P = 9,000 \)[/tex]
- [tex]\( A = 17,000 \)[/tex]
- [tex]\( r = 0.04647541 \)[/tex]
Substitute these values into the formula:
[tex]\[
t = \frac{\ln(\frac{17,000}{9,000})}{\ln(1 + 0.04647541)}
\][/tex]
6. Calculate [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(1.8888889)}{\ln(1.04647541)}
\][/tex]
Calculating the logarithms and dividing them gives:
[tex]\[
t \approx 14.00000116978684
\][/tex]
7. Round to the Nearest Whole Year:
The calculated time [tex]\( t \approx 14.00000116978684 \)[/tex] years.
When rounding to the nearest whole number, we get:
[tex]\[
\text{Number of years} = 14
\][/tex]
Therefore, it would take approximately 14 years for [tex]$9,000 to grow to $[/tex]17,000 with an annual rate of change of 4.647541%.
